Coupling and stability of interfacial waves in liquid metal batteries

We investigate the coupling dynamics of interfacial waves in liquid metal batteries and its effects on the battery's operation safety. Similar to aluminium reduction cells, liquid metal batteries can be highly susceptible to magnetohydrodynamically exited interfacial instabilities. The resulting waves are capable of provoking short-circuits. Owing to the presence of two metal-electrolyte interfaces that may step into resonance, the wave dynamics in liquid metal batteries is particularly complex. In the first part of this paper, we present a potential flow analysis of coupled gravity-capillary interfacial waves. While we are focusing here on liquid metal batteries with circular cross-section, the theory is applicable to arbitrary stably stratified three-layer systems. Analytical expressions for the amplitude ratio and the wave frequencies are derived. It is shown that the wave coupling can be completely described by two independent dimensionless parameters. We further provide a decoupling criterion that suggests that wave coupling will be present in most future liquid metal batteries. In the second part, the theory is validated by comparing it with multiphase direct numerical simulations. An accompanying parameter study is conducted to analyse the system stability for interfaces coupled to varying degrees. Three different coupling regimes are identified involving characteristic coupling dynamics. For strongly coupled interfaces we observe novel instabilities that may have beneficial effects on the operational safety.